Econometric Analysis of Productivity: Theoryand Implementation in R

Robin C. Sickles ∗ Wonho Song † Valentin Zelenyuk ‡§

September 3, 2018

Abstract

Our chapter details a wide variety of approaches used in estimat-ing productivity and efficiency based on methods developed to es-timate frontier production using Stochastic Frontier Analysis (SFA)and Data Envelopment Analysis (DEA). The estimators utilize panel,single cross section, and time series data sets. The R programs in-clude such approaches to estimate firm efficiency as the time invariantfixed effects, correlated random effects, and uncorrelated random ef-fects panel stochastic frontier estimators, time varying fixed effects,correlated random effects, and uncorrelated random effects estima-tors, semi-parametric efficient panel frontier estimators, factor modelsfor cross-sectional and time-varying efficiency, bootstrapping methodsto develop confidence intervals for index number-based productivityestimates and their decompositions, DEA and Free Disposable Hullestimators. The chapter provides the professional researcher, analyst,

∗Corresponding author. Professor of Economics, Department of Economics, Rice Uni-versity, 6100 Main Street, Houston, Texas 77005, USA; phone: +1 713-348-3322; email:rsickles@rice.edu

†Professor of Economics, School of Economics, Chung-Ang University, 84 Heukseok-ro, Dongjak-gu, Seoul 06974, Republic of Korea, Phone: +82-2-820-5493, email: wh-song@cau.ac.kr

‡Professor of Economics, School of Economics, University of Queensland, 530 ColinClark Building (39), St Lucia, Brisbane, Qld 4072, Australia; Phone:+ 61 7 3346 7054e-mail:v.zelenyuk@uq.edu.au

§The authors would like to thank Kerda Varaku for her assistance in preparing ourmanuscript. All remaining errors are the responsibility of the authors.

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Econometrics Using R., Handbook of Statistics, C.R. Rao and H.D. Vinod, eds., Elsevier, 41 (forthcoming, 2019) : 1-31.

statistician, and regulator with the most up to date efficiency modelingmethods in the easily accessible open source programming languageR.

Keywords: Production (technical) efficiency; Stochastic frontieranalysis; Data envelopment analysis; Panel data; Index numbers; Non-parametric analysis; Bootstrapping

1 Introduction

Our chapter provides a discussion of various statistical and mathematicalprocedures that firms, regulators, and academics and policy makers utilizein order to better understand performance and production (technical) effi-ciency of the entities they are benchmarking against other competitors orpeers. As well we discuss, give examples of, and provide extensive links toR programs that implement these various methods and approaches. Thevarious methods and approaches distinguish themselves by leveraging the es-timation of relative performance and of production efficiency measures onregression-based methods and on linear programming methods, the formerreferred to as Stochastic Frontier Analysis (SFA) and the latter as Data En-velopment Analysis (DEA). We discuss the main approaches in turn, theirrelative strengths and weaknesses, and briefly touch on ways to aggregatethe various methods using model averaging approaches.

Our chapter is organized in the following way. We first briefly discuss themotivation for using such methods that rest on the presence of production ef-ficiency differences among units of production that are being compared. Thevarious sets of rationales for such a discipline as efficiency and productivityanalysis can be categorized into three main groups of motivating factors.They are based on varying management practices, which deliver heteroge-neous outcomes, behavioral economics, and the presence of X-efficiency. Wethen discuss the two main classes of production efficiency estimators, the SFAand the DEA estimators. We next discuss examples of R programs that im-plement these various classical estimators and various extensions that havebeen introduced in the recent literature as well as protocols for accessing Rprograms on well-documented websites and other open source sites. We thenprovide concluding remarks.

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2 Why Estimate Production (Technical) Ef-

ficiency?

One of the most compelling rationales for the study of production efficiencyis substantial heterogeneity in management practices and resulting changesin the operating efficiency of a firm. A consensus in the empirical literatureexists that indicates substantial productivity differences both within a firm1

over time and among firms (Lieberman et al. (1990); Foster et al. (2008);Hsieh and Klenow (2009); Hall and Jones (1999)). Glaister (2014), amongothers, have noted that management practices are a key factor in explain-ing such productivity differences. Other factors, such as expenditures onR&D, utilization of capacity, and technology adoption, which are key deci-sions of management (Nallari and Bayraktar (2010)), are typically controlledby management practices. Bloom et al. (2012) regressed gross domestic prod-uct (GDP) per capita on a set of indicators of management practices among17 countries. These indicators of management practices explained 87% ofthe variation in per capita GDP. These findings are corroborated in a moremicro oriented study of Indian firms by Bloom et al. (2013). The engineer-ing mechanism, blueprints, formal structural statistical model, or economicmodel that explains how a firm’s productivity is linked to management skillsand practices has not been developed in a way that lends itself to empiri-cal analysis and to the generation of relative technical efficiency differencesamong peer firms. We thus tend to view such a factor as an unobservable la-tent factor and assume that management practices are one of the key factorsin firm productivity. Another factor is innovation but innovation is often-times facilitated by decisions and practices of management. The literature onthe effects of management practices on production efficiency, labor efficiency,and related measures of a firm’s financial success is quite dense.2 Behavioraleconomics provides another motivation for why firms may not operate at thefrontier of production efficiency. The assumptions of efficient markets andrational decision makers have been leveraged with substantial success by neo-classical economists, but the footing on which these assumptions rest may bea bit loose and slippery.

1We will use the term “firm” to denote any generic unit whose production efficiency isbeing measured and estimated.

2Much of this literature is summarized and referenced Grifell-Tatje and Lovell (2013,2015) and Grifell-Tatje et al. (2018).

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X-efficiency theory is a pragmatic paradigm that admits to the prevalenceof various sources on inefficiency in economics. It was introduced by Leiben-stein (1966, 1975, 1987), who pointed out that agency problems, asymmetricinformation, and monitoring by regulators were all factors that generatedincentives to engage in sub-optimal decision making, absent these factorsand constraints. Drawing from studies of the health care industry, telecom-munications, airlines, and education, Frantz (1997, 2007) have documentedthe ubiquitous existence of levels of inefficiency with the predictions from X-efficiency theory and have also concluded that such production inefficiency ismuch more significant than inefficiencies due to incorrect output and inputallocations (allocative inefficiency). Other studies that have documentedsuch inefficiency levels in the banking system can be found in Kwan (2006);Jiang et al. (2009); Fu and Heffernan (2009); Yao et al. (2008); Rezvanianet al. (2011); Bauer and Hancock (1993); Mester (1993); DeYoung (1998).And as emphasized by Frantz (1997)

“...what becomes of the word maximize if non-maximize is not possible?Is the concept of efficiency important if the possibility of inefficiency is ruledout a priori? The importance of efficiency remains as long as economicsremains important...”

X-efficiency theory and the methods that we present in our chapter isbased on an interdisciplinary approach that combines psychology, manage-ment, statistics, applied mathematics, and engineering.

Finally, the usefulness and importance of SFA and DEA methods havepassed the market test as they are required to be used in a wide varietyof regulatory decision-making settings in Europe and elsewhere (Bogetoft(2013), Agrell et al. (2017)).

3 Regression-Based Methods to Estimate Pro-

duction Efficiency

We begin our formal discussion of production efficiency measurement by firstdefining a few sets and functions that are necessary for our presentationof methods and how to implement them. To facilitate further discussion,

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let inputs and outputs be represented by x = (x1, ..., xN)′ ∈ RN+ and y =

(y1, ..., yM)′ ∈ RM+ , respectively and the production process characterized by

a technology set T that satisfies standard regularity conditions (Sickles andZelenyuk (2018)). The output set is defined as

P (x) ≡ {y ∈ <M+ : y is producible from x ∈ <N+}. (1)

The production function f : <N+ → <1+ is defined as f(x) ≡ max{y: y ∈

P (x)}. The maximum of the production function exists and is unique owingto the fact that P(x) is a compact set. Details on the regularity conditionsthat assure the maximum is unique and exists also can be found in Sicklesand Zelenyuk (2018). When there is more than one output one can usethe generalization of the production function, output orientated (Shephard(1970)) distance function Do : <N+ × <M+ → <1

+ ∪ {+∞} as Do(x, y) ≡inf{θ > 0 : y/θ ∈ P (x)}. For a technology with only one output and holdingthe inputs constant, Do is the ratio of actual output to potential (maximal)output. Then frontier production is f(xo) = yo/Do(x

o, yo). When thereare multiple-outputs, the Shephard output distance function is the smallestscalar required to radially expand all outputs to the output set’s boundary,again at a fixed level of inputs.

3.1 The Stochastic Frontier Paradigm

In cross-sectional Stochastic Frontier Analysis (SFA) there are at least twosources of error (panel extensions may extend the identifiable error compo-nents for four distinct sources of error). In addition to the error that isappended to the parametric or nonparametric production or distance func-tion to account for standard statistical noise (assumed to have mean zero) anadditional source of error is added to address the asymmetry in the errors dis-played by most production or distance function estimates that is due to theboundary property of the function being estimated. Such a one-sided erroris utilized additively as a placeholder for production or technical inefficiency,which diminishes the level of observed output from the frontier productionor distance function specified parametrically or nonparametrically. An ex-ample of such an SFA model for a linear in logs (Cobb-Douglas) productionfunction is,

ln(y) = β0 +N∑k=1

βk ln(xk) + ε, (2)

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ε = v − u. (3)

Here v is a random term with mean zero and variance σ2v . The error term

u represents the latent production inefficiency term. The latent productioninefficiency term has positive support with a mean µ > 0 and a variance σ2

u.3

Although these assumptions have been modified to account for more realisticempirical settings, this forms the basis for the canonical model used in SFA.Most of the literature in SFA has worked to generalize the distributions ofu and v, their moment properties, stochastic properties of the inputs (andfor multi-output distance functions the outputs as well), and different de-pendency patterns among the disturbances and the inputs and outputs, aswell as possibly distinguishing between variables that are used in developingthe production boundary and environmental or confounding factors that mayinfluence production efficiency, that is the deviations between the boundaryand non-boundary observations, as opposed to influencing the boundary perse.

3.1.1 Corrected OLS

One of the first approaches to develop measures of production efficiency isthe corrected ordinary least squares. The steps employed in such an exer-cise involve first estimating the production (or distance) function by ordinaryleast squares to get estimates of the average production relationship. In thesecond step one simply shifts the intercept (the example here is for the singleoutput production function) to ensure that the residuals are all non-positive(Olson et al. (1980)). With only mild assumptions, OLS will yield mini-mum variance unbiased linear estimators. The step with the Cobb-Douglasproduction function involves estimating

ln(yi) = β0 + β1 ln(xi1) + · · ·+ βN ln(xiN) + εi, i ∈ {1, ..., n}. (4)

The second step uses the intercept correction

βcols0 := β0 + maxi{εi}, i = 1, ..., n, (5)

where εi are OLS residuals. The last step estimates production efficiency fora firm as

Technical Inefficiency = εcolsi := maxi{εi} − εi, i ∈ {1, ..., n}. (6)

3Cost inefficiency can be modeled with a one-sided error with only negative support.

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Thus the corrected production function that estimates the production frontieris

ln(yi) = βcols0 + β1 ln(xi1) + · · ·+ βN ln(xiN)− εcolsi , i ∈ {1, ..., n}. (7)

3.1.2 Stochastic Frontier Model

Aigner et al. (1977) (hereafter ALS) and Meeusen and van den Broeck (1977)pursued a parametric model of the stochastic frontier based via maximumlikelihood. Average inefficiency is defined in terms of the performance of afirm to the firm identified as having the best-practices, as measured by itslevel of efficiency. The original ALS model used a half-normal distributionfor the efficiency term and a normal error for the idiosyncratic disturbance.Many other distributions have been considered, usually for the inefficiencyterm. These include the exponential, truncated normal, gamma, and dou-bly truncated normal (Stevenson (1980), Greene (1980a,b),Qian and Sickles(2008), Almanidis and Sickles (2012), and Almanidis et al. (2014)). In thecanonical SFA the composite error terms are assumed to be independent. Ifthe production function is linear in logs then a convenient parameterizationis

yi = f(xi|β) exp(εi), i = 1, · · · , n (8)

and for ε = v − u the inefficiency of firm i is measured as

exp(−ui) ≡yi

f(xi) exp(vi), i = 1, · · · , n. (9)

The model is usually specified after log-transforming the production re-lationship

ln yi = ln f(xi|β) + vi − ui, i = 1, · · · , n (10)

and assumingvi ∼ N (0, σ2

v)

andui ∼ |N (0, σ2

u)|

and that ui and vi are independent and i.i.d., it follows that the density of εiis:

fεi(ε) =2

σφ( εσ

)[1− Φ

(ελ

σ

)], −∞ ≤ ε ≤ +∞ (11)

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where Φ(·) is the standard normal distribution function, σ2 = (σ2v +σ2

u) , andλ = σu/σv. The composite errors, ε1, · · · , εn, are of course not observed, butfrom our model we know that

εi = ln yi − ln f(xi), i = 1, · · · , n. (12)

The log-likelihood function is

`(y1, ..., yn|β, λ, σ2) =n

2ln( 2

π

)− n

2ln(σ2)− 1

2σ2

n∑i=1

[ln yi − ln f(xi)

]2+

n∑i=1

ln[1− Φ

( [ln y − ln f(xi)]λ

σ

)].

(13)Inefficiency has a mean and variance given by

E(ui) ≡ µ =

√2√πσu (14)

and

V (ui) =

(π − 2

π

)σ2u (15)

and thus the composed error’s mean and variance is

E(εi) = E(vi − ui) = E(−ui) = −µ = −√

2√πσu (16)

V (εi) = V (vi − ui) = V (v) + V (u) = σ2v +

(π − 2

π

)σ2u (17)

andcov(εi, εj) = 0,∀i 6= j. (18)

Estimation of individual inefficiencies Cross-sectional SFA, unfortu-nately, does not provide a consistent estimate of the efficiency of each firm.A simple solution by Materov (1981) and further developed in Jondrow et al.(1982) uses E(ui|εi) as a point estimate of ui. This estimator has its draw-back as it is contaminated by statistical noise but it is appealing in that,from the law of iterated expectations,

E(ui) = Eε[E(ui|εi)] (19)

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and thus to develop a consistent estimator of E(ui|εi) we can write

E(ui|εi) ≡∞∫0

ufui|εi(u|ε)du (20)

where, from the definition of the conditional density,

fui|εi(u|ε) ≡fui,εi(u, ε)

fεi(ε). (21)

Based on the normal/half normal composed error structure typically used inSFA we can write

E(ui|εi) = µ∗ + σ∗1

1− Φ(−µ∗/σ∗)φ

(−µ∗σ∗

)(22)

or

E(ui|εi) = σ∗

[µ∗σ∗

+1

1− Φ(−µ∗/σ∗)φ

(−µ∗σ∗

)]. (23)

Using the parameterization −µ∗σ∗

= σ2uε

σ2 σvσuσ

= σuεσσv

= ελσ

, the conditional expec-

tation is:

E(ui|εi) =σvσuσ

[−εiλσ

+φ (εiλ/σ)

1− Φ(εiλ/σ)

](24)

for which a consistent estimate can be obtained based on consistent estimatesof the model parameters (e.g., obtained via MLE or COLS, or the ModifiedOLS procedures of Olson et al. (1980)). Thus a (conditional) consistentestimator of each firm’s efficiency level can be based on this last equation.

3.1.3 Panel Stochastic Production Frontiers

The cross-sectional stochastic frontier model has a number of drawbacks thathave been addressed over the forty years since it was introduced. As wepointed out, a consistent estimate for a firm’s technical efficiency is notavailable, only a consistent estimate of the conditional mean of the firm’sefficiency level. Of course the canonical ALS model is fully parametric andinputs are assumed to be uncorrelated with the regressors, which is partic-ularly troubling when the regressors are inputs and the latent factor thattypically is assumed to account for inefficiency is unobservable managerialexpertise, an input whose independence from the levels of capital and labor

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used in production is a questionable assumption, and of course if a firm isaware of its level of technical efficiency this information, although unknownto the analyst, should not be independent of the firm’s input choices.

These problems are potentially avoidable if one has panel data (Pitt andLee (1981); Schmidt and Sickles (1984)), although the possible endogeneityof input choice must be addressed with care and its treatment using fixedeffects type estimators may not be completely satisfactory. We will returnto this issue shortly. If we have a panel of firms, then if the unobservedfirm effects represent technical efficiency they can be estimated consistentlyfor large T (assuming that our sample of firms n is large) after controllingfor inputs and other environmental and observable factors that may impactproduction. To show how this is accomplished with panel data we use thegeneral treatment discussed in Schmidt and Sickles (1984) (SS), which alsoconsiders the Pitt and Lee (1981) parametric random effects model. Themodel is

yit = α + x′itβ + vit − ui, i = 1, ..., n; t = 1, ..., T, (25)

which can be rewritten as

yit = α∗ + x′itβ + vit − u∗i (26)

where α∗ = α − µ; u∗i = ui − µ; E(ui) = µ ≥ 0, and where xit is a vector ofN inputs. If we let αi = α∗ − u∗i then the model becomes the usual paneldata model

yit = αi + x′itβ + vit (27)

and cross-sectional effects that can be viewed as random, fixed, or simply ig-nored. Five estimators of the classical panel data model with time-invarianteffects are discussed in SS. These are the pooled OLS model, the fixed effectswithin estimator, the random effects model, the Hausman-Taylor estimator,and the fully parametric random effects MLE model (Pitt and Lee (1981)).Schmidt and Sickles (1984) discuss the asymptotics of each of these estima-tors and the assumptions needed in order for the parameter estimates andestimates of the technical efficiency level to be consistently estimated.

Technical efficiency effects estimates with the SS estimators do not changeover time and this is a strong and often unreasonable assumption that is notnecessary. The suite of panel stochastic frontier estimators developed by

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Cornwell et al. (1990) (CSS) had a parameterization that allowed for time-varying heterogeneity and is based on the model:

yit = x′itβ + z′iγ + w′itδi + vit i = 1, · · · , n; t = 1, · · · , T, (28)

where xit, zi and wit are N ×1, J×1 and L×1 vectors, respectively, and theparameter vectors β, γ and δi are dimensioned conformably. With δ0 = E[δi],and δi = δ0 + ui the model can be written as:

yit = x′itβ + z′iγ + w′itδ0 + εit,

where

εit = vit + w′itui (29)

and where ui is assumed to be i.i.d., zero mean random variables with co-variance matrix ∆.The error term vit is assumed to be i.i.d., with zero meanand constant variance σ2

v . The basic model assumes that vit is uncorrelatedwith z, x, and ui. In order to allow for time-varying technical efficiency thestochastic frontier model of Schmidt and Sickles (1984)

yit = α + x′itβ + vit − ui = αi + x′itβ + vit. (30)

can be modified by replacing the αi with, e.g.,

αit = θi1 + θi2t+ θi3t2, (31)

and the model in matrix form becomes

y = Xβ + Zγ +Wδ0 + ε, (32)

ε = Qu+ v. (33)

Details of the estimator can be found in Cornwell et al. (1990). Fixedeffects, random effects, and Hausman-Taylor estimators of the CSS modelwere developed to estimate productivity efficiency that is time varying andallow for consistent estimation under large n and T asymptotics of the timevarying productivity efficiency for each firm. The parameter δi is estimatedby regressing the residuals (yit − x

′itβ) for firm i on wit. This amounts to

regressing the within residual on a constant term, time, and time-squared for

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the specification we introduced for αit above and relative efficiencies can beapproximated (in the linear in logs production function) by

αt = maxi

(αit) i = 1, ..., n (34)

uit = αt − αit. (35)

Parametric MLE case also be used to estimate a random effects timevarying technical efficiency models. Kumbhakar (1990) and Battese andCoelli (1992) present two such models. In the former inefficiency is modeledas

uit = (1 + exp(bt+ ct2))−1τi, (36)

where a and b are parameters to be estimated and where τi’s distribution isassumed to be i.i.d N+(0, σ2

τ ) and vit is i.i.d N (0, σ2v). In the latter specifi-

cation inefficiency is modeled as

uit = ηtτi = {exp[−η(t− T )]}τi, t = 1, . . . , T ; i = 1, . . . , n, (37)

where vit is i.i.d N (0, σ2v) random variables, τi is assumed to be i.i.d. and has

a non-negative truncated distribution N (µ, σ2), and η is a scalar parameter.

3.1.4 Second-and Third-Generation Stochastic Frontier Models

The models we have just discussed provide the basic statistical intuition forSFA. Much has been done to extend and generalize the canonical models wehave discussed and space prohibits us from providing comparable detail onthese second and third generation extensions. However, the R codes that wewill discuss shortly are equipped to deal with a number of these relativelynew modeling scenarios and so we give a brief summary of what these secondand third generation SFA models deliver and the general ideas behind howthey are formulated and specified. We leave it to the reader to seek out theoriginal sources and recent books, handbooks, and survey articles that wehave referenced and that provide more detailed treatments.

Researchers have provided further generalizations of the CSS time-varyingtechnical efficiency model that focus on two main aspects of the model. Thefirst is to allow for a factor-type structure for the cross-section and timevarying technical efficiency term (these are almost all exclusively panel data

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extensions). The second is to decompose the time invariant and time varyingefficiency terms into a portion that is skewed, to represent technical ineffi-ciency, and a portion that is symmetric, the latter somewhat courageouslycalled the “true” effects model. There are four other generalizations thatsome may view as comparable to these but there is simply not enough spaceto address them in any substantive way. We will mention them and pro-vide several references as well as links to R codes that deal with these fouradditional issues, which are: environmental factors, endogeneity, Bayesianmethods for SFA, and nonparametric specifications of the technology and ofthe error structure.

We briefly discuss these sets of generalizations and available R code toimplement them.

Factor Models and SFALee (1991) and Lee and Schmidt (1993) were the first to propose a (one

component) factor model to address time-varying and cross-sectional specificproduction efficiency. Their model is

yit = αt + x′itβ + vit − uit for i = 1, . . . , n; t = 1, . . . , T, (38)

or as

yit = x′itβ + αit + vit, (39)

where αit = αt − uit is the time varying cross-sectional specific technicalefficiency term, which are modeled as s as

αit = ηtδi. (40)

Here ηt, t = 1, ..., T, are the time-varying effects to be estimated and δi, i =1, ..., n, are the firm effects. Lee and Schmidt (1993) provide fixed-effectsand random effects estimators for this one-factor model based one nonlinearregression estimators.

Ahn et al. (2007) generalized this model to allow for multiple factors thatchange over time and specify the production frontier as

yit = δt + x′itβ + vit − uit (41)

= x′itβ + ηit + vit, i = 1, 2, . . . , n, t = 1, 2, . . . , T, (42)

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where, again, vit is the usual disturbance term and uit ≥ 0 is the inefficiencyterm and where ηit ≡ δt − uit is the time-varying and cross-sectional spe-cific technical efficiency term that is expressed as a linear combination of punrestricted components,

ηit = θ1tα1i + θ2tα2i + · · ·+ θptαpi =

p∑j=1

θjtαji. (43)

The model is estimated using generalized methods of moments. Ahn et al.(2013) provide a focused study of consistency properties of their Ahn et al.(2007) model when different sorts of dependency relationships exist betweenthe production efficiency effects and the regressors.

In the Ahn et al. (2007) model the effects are multiplicative. Kneip et al.(2012) (KSS) provided a more general model than Ahn et al. (2007) by al-lowing for a general nonparametric time-varying and cross-section specificproductivity efficiency. The KSS model is:

yit = β0(t) +N∑j=1

βjxitj + ui(t) + vit, i = 1, . . . , n; t = 1, . . . , T. (44)

The ui(t)’s are assumed to be smooth time-varying individual effects thatsatisfy a normalization that

∑i ui(t) = 0 and are a linear combination of

L < T basis functions (common factors) g1, ..., gL:

ui(t) =L∑r=1

θirgr(t). (45)

The term β0(t) is an average function that is eliminated by centering themodel yielding

yit−yt =N∑j=1

βj(xitj−xtj)+ui(t)+vit−vi, i = 1, . . . , n; t = 1, . . . , T (46)

where yt = 1n

∑i yit, xtj = 1

n

∑i xitj and vi = 1

n

∑i vit. As with the CSS

and many other panel stochastic frontier estimators, technical efficiency iscalculated as TEi(t) = exp{ui(t)−maxj=1,...,n(uj(t))}, just as it is calculatedwith the CSS estimator. The estimator used is a three step one that first esti-mates β using penalized least squares and smoothing splines to approximate

14

the factors, then second generates estimates of the covariance structure of theui(t)

′s , and the third step estimates the basis functions gr and then updatesestimates of ui by

∑Lr=1θirgr. The KSS estimator and related estimators

such as those introduced by Bai and Ng (2002) and Bai (2009), as well astests for the number of factors, have been programmed in R and are availablenot only in the suite of programs we discuss at the end of this chapter butalso can be found on the website referenced in Bada and Liebl (2014).

True Fixed Effects and SFAGreene (2005a,b) proposed a stochastic panel frontier model in which the

intercept fixed effects were not measures of persistent inefficiencies, as hadbeen assumed by SS, but rather was simply firm specific heterogeneity. Thishas become known as the“true”fixed effects model, although a less ambitiouslabel may be the panel frontier with only transitory inefficiency. The modelis

yit = αi + x′itβ + vit − uit (47)

εit = vit − uit, (48)

or

yit = αi + x′itβ + εit (49)

and thus the source of inefficiency is uit, which is distinguished from the in-tercept terms αi. Greene outlines estimators for both the fixed effects andrandom effects panel stochastic frontier model with only transitory ineffi-ciency based on simulated MLE and technical efficiency estimates are basedon the familiar expression

TEit = exp[−{max(uit)− uit}]. (50)

Colombi et al. (2011), Colombi et al. (2014), and Tsionas and Kumbhakar(2014), provided further generalizations of this model and alternative estima-tors based on Bayesian methods, full information maximum likelihood, quasi-MLE, and method of moments. One important generalization proposed bythese authors specifies the persistent and transitory efficiency terms as sepa-rate skew-normal errors based on the following parameterization ofColombiet al. (2014)

yit = x′itβ + αi + vit − uit − ηi, (51)

15

where εit = αi+vit−uit−ηi is the error structure and where uit and ηi are non-negative random variables that capture firm-specific random effects, randomnoise, short-run technical inefficiency, and long-run technical inefficiency.

Environmental factors can be introduced into the SFA paradigm in anatural way by including them in a vector of variables that is assumed toimpact the mean and the variance of production efficiency. For an excellenttreatment of the problem see Wang and Schmidt (2002), Simar and Wilson(2008) and Kim and Schmidt (2008). The marginal effects of the environ-mental factors, based on Battese and Coelli (1995), can be estimated usingthe R package Frontier (http://cran.r-project.org/web/packages/frontier/)developed by Arne Henningsen.

Various dependency structures have been assessed within the SFA paradigm.An excellent source of recent work on this topic can be found in Kumbhakarand Schmidt (2016).

No one has had more of an impact on the introduction of Bayesian meth-ods into the SFA and DEA paradigms than Mark Steel and Mike Tsionas(Griffin and Steel (2007), Tsionas and Papadakis (2010), and Liu et al.(2017)). The WinBUGS package of Steele is Fortran and Matlab-based opensource code that can be modified to run in R. Nonparametric models in SFAhave been studied in a variety of settings by Park et al. (1998), Adams et al.(1999), Park et al. (2003), Park et al. (2007), among others, and the R codesfor the latter models are discussed in more detail in the last sections of thischapter on specific implementation of open source R codes for SFA and DEA.Experimental R codes that can estimate both parametric and non-parametricstochastic frontiers can also be found in the experimental R-Forge packagemaintained by Arne Henningsen.

4 Envelopment Estimators

In this section we will briefly review another very popular approach in themeasurement and empirical estimation of the efficiency of economic systems(firms, industries, etc.), called Data Envelopment Analysis (DEA).

4.1 The Origins of DEA

In a nutshell, DEA is rooted in and coherent with theoretical economic mod-eling using Activity Analysis Models (AAM) and is estimated via the powerful

16

linear programming approach. Its name was branded in the seminal work ofCharnes et al. (1978), who refined and generalized the approach of Farrell(1957) to estimate production efficiency, which in turn was influenced byseminal works of Debreu (1951), Koopmans (1951a,b) and Shephard (1953,1970).

4.2 The Basic DEA Model

In his seminal work, Farrell (1957) focused on the constant returns to scalemodel, with multiple inputs and a single output. About two decades later,Charnes et al. (1978) generalized Farrell’s approach to the multi-output casebut started with a very different formulation–a fractional programming prob-lem formulation with the objective being to optimize the ratio of a weightedaggregate of outputs to a weighted aggregate of inputs (i.e., a kind of produc-tivity index). They then transformed this problem into a linear programming(LP) problem and derived its dual, which turned out to be the (generalizedversion of) AAM proposed by Farrell. Specifically, the formulation of Charneset al. (1978) for estimating the efficiency score of a firm or decision-makingunit (DMU) j ∈ (1, ..., n) with an allocation (xj, yj), states

Eji,CCR = max

v1,...,vN ;u1,...,uM

{ ∑Mm=1 umy

jm∑N

l=1 vlxjl

:∑Mm=1 umy

km∑N

l=1 vlxkl

≤ 1, k = 1, ..., n,

um ≥ 0, vl ≥ 0, l = 1, ..., N ;m = 1, ...,M

}(52)

where u′ = (u1, ..., uM) and v′ = (v1, ..., vN) are optimization variables (alsocalled here ‘multipliers’). This DEA formulation is more popular in theoperations research and management science literature and is often referredto as the ‘CCR model’ or the ‘multiplier form of DEA’ under CRS, additivityand free disposability.

Importantly, Charnes et al. (1978) then showed that (52) is equivalent tothe AAM version of the DEA-estimator of the Farrell input oriented technicalefficiency score of a DMU with an allocation (xj, yj) under the assumptionof CRS (also assuming additivity and free disposability of all inputs and alloutputs), formulated as

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Ei(xj, yj) ≡ min

λ,z1,...,znλ (53)

s.t.n∑k=1

zkykm ≥ yjm,m = 1, ...,M, (54)

n∑k=1

zkxkl ≤ λxjl , l = 1, ..., N, (55)

λ ≥ 0, zk ≥ 0, k = 1, ..., n, (56)

which is an LP problem that can be solved via any standard LP solver.This formulation is more common in the economics literature (largely dueto its connection to works of Debreu (1951) and Koopmans (1951a,b)), andis often referred to as the envelopment form of DEA under CRS, additivityand free disposability. We will use this envelopment formulation to describeother variants of DEA, though it is useful to keep in mind that there isalso a multiplier form that optimizes what can be viewed as a normalizedproductivity index.

The previous formulation looks at minimization of all inputs, while keep-ing outputs fixed, and hence is called the input orientation. Similarly, theDEA-estimator of the Farrell output oriented technical efficiency score ofany (xj, yj) allocation, under the assumptions of CRS, and additivity andfree disposability of all outputs and all inputs is formulated as

Eo(xj, yj) ≡ max

λ,z1,...,znλ (57)

s.t.n∑k=1

zkykm ≥ λyjm,m = 1, ...,M, (58)

n∑k=1

zkxkl ≤ xjl , l = 1, ..., N, (59)

λ ≥ 0, zk ≥ 0, k = 1, ..., n. (60)

Immediately note that for these formulations we haveEi(x

j, yj) = 1/Eo(xj, yj) for any (x, y), as is required theoretically due to

CRS.

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Also note that the reciprocals of these estimated Farrell efficiency mea-sures also give estimates of the input and output oriented Shephard’s distancefunctions (Shephard (1953, 1970)), under the same assumptions on technol-ogy, i.e., CRS, additivity and free disposability.

Sometimes a researcher may not be interested in fixing only the levelsof inputs or outputs, but may want to simultaneously expand outputs andcontract inputs, thus requiring other orientations. The DEA-estimator of ageneral efficiency measure of any (xj, yj) allocation in such a case (also hereunder the assumptions of CRS and additivity and free disposability of alloutputs and all inputs), can be obtained as follows

GEo(xj, yj) ≡ max

λ1,...,λN ,,θ1,...,θM ,z1,...,znf(λ1, ..., λN , , θ1, ..., θM) (61)

s.t.n∑k=1

zkykm ≥ θmyjm,m = 1, ...,M, (62)

n∑k=1

zkxkl ≤ xjl /λl, l = 1, ..., N, (63)

λ ≥ 0, zk ≥ 0, k = 1, ..., n, (64)

where certain restrictions can be imposed on the objective functionf(λ1, ..., λN , , θ1, ..., θM) and its arguments, depending on the interest of theresearcher. For example, restricting f to be additive (but summing only pos-itive arguments) will give the DEA estimate of the general Russell efficiencymeasure and an additional restriction of θ1 = ... = θM = 1 will turn it intothe input oriented Russell efficiency measure (as was originally introducedby Fare and Lovell (1978)). If one instead restricts λ1 = ... = λN = 1then one obtains the output oriented Russell efficiency measure. Mean-while, restricting f(λ1, ..., λN , θ1, ..., θM) to be multiplicative (a geometricmean) will generate the multiplicative-Russell efficiency measure introducedby Fare et al. (2007). Furthermore, if instead one imposes θ1 = ... = θM = θand λ1 = ... = λN = λ then the DEA estimate of the what has been termedthe general hyperbolic efficiency measure is obtained. This latter measuresometimes also appears with the additional restriction that θ = λ, which im-poses the properties of equiproportional expansion (contraction) of all output(inputs). Note that in general, this last formulation is no longer an LP prob-lem and therefore, it is typically more challenging to estimate.

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Another very general measure, as well as a primal characterization oftechnology, is the directional distance function,4, which can be estimated viathe following DEA formulation (here also under the assumptions of CRS andadditivity and free disposability of all outputs and all inputs):

Dd(xj, yj|dx, dy) ≡ max

λ,z1,...,znλ, (65)

s.t.n∑k=1

zkykm ≥ yjm + λdym ,m = 1, ...,M, (66)

n∑k=1

zkxkl ≤ xjl − λdxl , l = 1, ..., N, (67)

λ ≥ 0, zk ≥ 0, k = 1, ..., n. (68)

In a similar fashion, DEA can be used to model and estimate cost, revenueand profit functions and associated efficiency measures. More details on thiscan be found in Sickles and Zelenyuk (2018).

4.3 The Myriad of DEA models

The approach briefly described in the previous section in the context of differ-ent efficiency measures constitutes the canonical forms of the DEA paradigm.Essentially, all the other versions are modifications or extensions of the mod-els we have outlined. In this section we briefly discuss a few of these modi-fications and extensions.

Oftentimes, extensions are obtained by imposing various additional con-straints onto either the envelopment form or the multiplier form of DEA,with an aim to better mimic the particular actual production process understudy. Such additional restrictions should be imposed with care, since theymay (and often do) affect other desirable properties related to previouslyadded constraints, such as CRS, convexity, free disposability, additivity, etc.

Additional constraints also may create computational complications, e.g.,turning the problem from a linear to a non-linear one or a hybrid problem

4The origins of ideas for this function go back to at least Allais (1943), Diewert (1983),and Luenberger (1992), and were later revived and thoroughly developed by Chamberset al. (1996)

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that may require integer-programming problems. This may result in possiblymaking the problem much harder to compute, possibly infeasible, or may giveinferior local optima or degenerate solutions.

4.3.1 Relaxing Constant Returns to Scale and Convexity

The first wave of extensions of the DEA-CRS model mainly focused on re-laxing assumptions such as CRS and convexity. This line of research waspursued by a number of researchers who extended and enriched the DEAparadigm, among them Afriat (1972); Fare et al. (1983); Banker et al. (1984);Deprins et al. (1984) and Petersen (1990); Bogetoft (1996), to mention a few.

Out of the many modifications and extensions of the canonical DEA prob-lem the several that have sustained the test of time and popularity are theDEA-VRS (variable returns to scale) and the DEA-NIRS (non-increasing re-turns to scale) models, which simply amount to adding additional constraintsin the form of

∑nk=1 zk = 1 or

∑nk=1 zk ≤ 1, respectively, to the DEA-CRS

formulations as described above.Meanwhile, the Free Disposal Hull (FDH) approach, can be implemented

via a hybrid of the linear programming and the integer programming prob-lems, which is formulated in exactly the same way as the DEA-VRS program-ming problem with the exception that the constraints “zk ≥ 0, k = 1, ..., n”are replaced with “zk ∈ {0, 1}, k = 1, ..., n”. For example, using the outputoriented Farrell efficiency with an allocation (xj, yj) the FDH formulation isgiven by

Eo(xj, yj) ≡ max

θθ (69)

s.t.n∑k=1

zkykm ≥ θyjm,m = 1, ...,M, (70)

n∑k=1

zkxkl ≤ xjl , l = 1, ..., N, (71)

n∑k=1

zk = 1, (72)

θ ≥ 0, zk ∈ {0, 1}, k = 1, ..., n. (73)

The value of such formulation is that it hints at the relationship betweenFDH and DEA-VRS: indeed, the DEA-VRS estimated technology set is sim-

21

ply the convex closure of the FDH-estimated technology set. Thus, the FDHestimator can be viewed as a special case of the data envelopment analy-sis approach since it also envelopes the data but does so without imposingconvexity. For historical reasons these names are kept separate to avoidconfusion. An alternative yet equivalent form of the FDH estimator can begiven via the min-max problem, which is faster to compute (e.g., see Simarand Wilson (2013) for more details).

Finally, stochastic versions of DEA and FDH are also available (e.g.,Simar (2007); Simar and Zelenyuk (2011)).

4.3.2 Modeling with Undesirable Outputs or with Congesting In-puts

Another important stream of DEA literature has focused on estimating tech-nologies with weak disposability of inputs or (and especially) outputs, toaccount for the fact that some outputs are undesirable (‘bad’) and someinputs can cause congestion.

The ideas for such modeling approaches go back to at least Shephard(1974), and then was elaborated on in Fare and Svensson (1980); Fare andGrosskopf (1983); Grosskopf (1986); Tyteca (1996); Chung et al. (1997),which defined the mainstream approach on this matter. More recently thismainstream approach was re-evaluated in several important works, includ-ing Seiford and Zhu (2002), Fare and Grosskopf (2003, 2004, 2009), Før-sund (2009), Podinovski and Kuosmanen (2011), Pham and Zelenyuk (2018,2017).5

While many proposals have been made, the most popular approach inthis context so far continues to be the mainstream one. In this approach, forexample, if one is interested in measuring the radial expansion of the goodoutputs (g) while having no more inputs (x) and no more of bad outputs (b),then the DEA estimate of the good-output-oriented Farrell technical efficiency

5Also see Dakpo et al. (2017) and Sueyoshi et al. (2017) for reviews of this researchstream.

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(under VRS) is given by

TEg(x, g, b) ≡ maxγ,z1,...,zn,δ

γ (74)

x ≥n∑k=1

zkxk (75)

gγ ≤ δn∑k=1

zkgk, γ ≥ 1 (76)

b = δn∑k=1

zkbk, 0 ≤ δ ≤ 1 (77)

n∑k=1

zk = 1, zk ≥ 0, k = 1, ..., n. (78)

For more theoretical and practical (computational) discussions on thistopic see Pham and Zelenyuk (2017) and references therein.

4.3.3 Other Streams of DEA

Another stream of DEA focuses on accounting for the network structure ofproduction technologies whether static or dynamic. This stream originatedin the seminal works of Fare and Grosskopf (1996); Fare et al. (1996) andwas taken further in many other works, e.g., see Kao (2009a,b, 2014) andreferences therein.

Another important stream of DEA literature is on the topic of weight re-strictions in multiplier form of DEA and the classical works here are by Dysonand Thanassoulis (1988); Charnes et al. (1990); Thompson et al. (1990), withmore recent and fundamental contributions including new interpretations (as‘technological trade-offs’) of various weight restrictions in DEA from Podi-novski and Bouzdine-Chameeva (2013), to mention just a few. Also seereviews on this topic by Allen et al. (1997); Podinovski (2015).

Yet another interesting research stream of DEA overlaps with game the-ory, which also has its roots in the seminal work of von Neumann (1945) andmore explicit treatment, for example, in Hao et al. (2000); Nakabayashi andTone (2006); Liang et al. (2008); Lozano (2012).

23

4.4 Statistical analysis of DEA and FDH

Another key research wave that brought DEA and FDH to a totally differentlevel is related to their statistical aspects—this wave was mainly influencedby the seminal works of Leopold Simar and many of his co-authors.

The first breakthrough in this area was made by Banker (1993), where thefirst proof of consistency of the DEA estimator was sketched for the single-output case (in output oriented context), and was pointed out that it belongsto the class of maximum likelihood estimators. This important discovery wasthen substantially enriched by Korostelev et al. (1995a,b) who proved con-vergence of the estimated technology to the true technology for both DEAand FDH estimators, and derived convergence rates of these estimators, clar-ifying that they depend on the dimension of the production model, yet alsohave some optimality properties under certain conditions.

The convergence properties for the multi-input-multi-output case werefirst presented in the seminal work of Kneip et al. (1998). Meanwhile, thediscovery of the limiting distribution of the DEA estimator was done by Gi-jbels et al. (1999), only for the 1-input-1-output case and a decade later,Kneip et al. (2008) derived it for the fully multivariate case for DEA withVRS and also proved consistency of various bootstrap procedures. The lim-iting distribution for the case of DEA with CRS was established by Parket al. (2010) and the limiting distribution of the FDH estimator for the fullymultivariate case was also derived by Park et al. (2000), while consistencyof the bootstrap for FDH estimator was first presented in Jeong and Simar(2006).

This stream also includes the approach of analyzing the DEA (or FDH)estimated efficiency scores. Perhaps the most popular of these is the so-called ‘two-stage DEA’, which involves regression analysis of efficiency scoreson some factors. The state of the art here is the approach proposed by Simarand Wilson (2007), which is based on truncated regression where the inferenceis done with the help of a double bootstrap.6 A non-parametric version ofthis approach (based on non-parametric truncated regression) was proposedby Park et al. (2008). Furthermore, methods to analyze the distributions ofDEA and FDH efficiency scores were explored in Simar and Zelenyuk (2006),while methods to analyze industry efficiency were explored by Simar andZelenyuk (2007). These approaches were applied in various contexts andindustries, e.g., Zelenyuk and Zheka (2006), Demchuk and Zelenyuk (2009),

6Also see Simar and Wilson (2011) for the discussion on caveats and limitations.

24

Curi et al. (2015), Chowdhury and Zelenyuk (2016), Du et al. (2018), tomention a few.

A particularly notorious drawback of DEA and FDH—not allowing fornoise and sensitivity to ‘super-efficient’ outliers—was addressed by Simar(2007); Simar and Zelenyuk (2011), who proposed their version of StochasticDEA and Stochastic FDH, which consists of two stages: (i) filter the datafrom the noise using a non-parametric stochastic frontier method7 and then(ii) use DEA or FDH on the filtered data.

More recently, Kneip et al. (2015) derived new central limit theoremsfor the context where DEA or FDH estimates are used in place of the trueefficiency and thus provided the foundation for many useful statistical testsinvolving DEA or FDH estimators, including the two-stage ‘DEA+regression’context. This foundation was then used by Kneip et al. (2016); Daraio et al.(2017b) to develop various statistical tests and by Simar and Zelenyuk (2017)to develop two new central limit theorems for the aggregate efficiency scores(industry efficiency, etc.) of the type described above and more work contin-ues in this area.

5 SFA Efficiency Software in R

Reproducing the R code needed to implement the methods we have discussedin our chapter is not feasible and thus we provide a short tutorial on how touse a suite of estimators that can easily be accessed via the website “Produc-tivity in R”that can be found as https://sites.google.com/site/productivityinr/.We use standard notations in the codes and let n be the number of firms,T the number of time series and nT the total number of observations, i.e.,nT = n× T . The data input files configured with the first column a nT × 1column vector of the dependent variable y, and the next k containing nT × 1vectors of the independent variables contained in x′. The convention used isfor the first T observations to be for the first cross-section, the second T ob-servations for the second cross-section and so on. The default program output(‘results.out’ is the default output filename) contains parameter estimates,standard errors, t-values, average technical efficiency, correlation of effectsand efficiencies, Spearman rank order correlation of effects and efficiencies,

7While originally Simar and Zelenyuk (2011) considered the approach of Kumbhakaret al. (2007) for the first stage, one could use other non-parametric SFA approaches, e.g.a more general version proposed by Park et al. (2015) or Simar et al. (2017).

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R-squared and adjusted R-squared.

5.1 Basic Model Setup

The default model is the panel data model we introduced in section 3, whichwe rewrite here:

yit = α + x′itβ − uit + vit, (79)

where the global mean (α) is subtracted out in the regression and where thisdemeaning method is applied to the suite of estimators discussed in SS andCSS as well as for the Battese and Coelli (1992) (BC) estimator. Timetrends can be added as well to account for disembodied technical changethat is available for adoption by all firms and various right-hand-side (rhs)variables that may be correlated with the effects can be identified in the mainR file. Efficiencies can also be averaged over different estimators using modelaveraging weights based on a simple average (AVE=0), AIC weights (AVE=1),or BIC weights (AVE=2). Outliers can be addressed by trimming (this doesnot apply to the BC and the DEA estimators).

5.2 Figures and Tables

Figures and tables are selected by setting values of 1 to print and 0 to skipthe results. There are a number of figures that are already set up to print.For example, when fig1=1 a figure for the average of efficiencies of the time-variant estimators is printed. When fig2=1 one for the efficiencies from thetime-variant estimators is printed. When fig3=1 a figure for the average ofefficiencies from all estimators is printed, while fig4=1 the weighted averageof the efficiencies is printed. tab1=1 prints a table for the average of efficien-cies from the time-variant estimators, tab2=1 prints one for the efficienciesfrom the time-variant estimators are printed and tab3=1 prints a table ofthe individual effects from all estimators. Also, setting firmeff=1 savesthe efficiencies of each individual firm for each estimator utilized.

5.3 Different Estimators

The R codes for the SFA models include the fixed effect, random effects, andthe Hausman-Taylor (HT) version of the stochastic panel frontier models ofSchmidt and Sickles (1984). The FR and the HT global parameters are set

26

to determine which model is estimated. Since the HT estimator of the panelstochastic frontier allows for selected regressors to be correlated with the ef-ficiency effects term the global option k1 is set at the number of variablesin X not correlated with the efficiency effects and are loaded into the firstk1 columns for the regressors. These different models are referred to as FIX,RND, and HT. The global options PSS1, PSS2, PSS3 designate the estimatorsfor the Park, Sickles and Simar (1998, 2003, 2007) models. The value of 1prints the results and 0 skips the results. Bandwidth selection is based onleave-one-out least squares cross-validation. The CSS global option allows forthe estimates of the Cornwell et al. (1990) models to be generated. A globaloption of CSS=1 prints results of the Fixed Effects/Within estimator (CSSW)estimator, while CSS=2 generated the GLS random effects estimator. Theefficient IV estimator (analogous to the HT estimator for the SS Model) isengaged with CSS=3. Finally, CSS=4 prints out all four model results. Timeinvariant variables should be placed at the end of X. The parameter zp isset equal to the number of time invariant variables. The Kneip, Sickles andSong (2012) factor-model is called by setting the global parameter KSS=1.

The KSS estimator also finds the number of factors and the code varies thenumber of factors from Lmax to Lmin and finds the first highest number atwhich the dimensionality test is not rejected. Cross-validation is used toestimate the optimal smoothing parameter. gr_st is the starting point ofthe grid search, gr_in is the increment in the grid search, and gr_en is theend point of the grid search. The Battese and Coelli (1992) model is calledby setting the BC option=1. The bounded inefficiency model of Almanidis,Qian, and Sickles (2014) is called by setting the BIE option=1. Differentdistributions for the lower bound on inefficiency can also be specified usingthe bie_dist parameter. For a bie_dist=0 the truncated is specified,while bie_dist=1 uses the truncated half normal, and bie_dist=2 the dou-bly truncated normal distribution. Finally, the Kutlu (2018) endogeneitycorrection for selected regressors (e.g., input levels in the production fron-tier) based on the CSS estimator is also available on this website along withinstructions for its use.

6 DEA Efficiency Software in R

The Jeon and Sickles (2004) model is the directional distance function methodoutlined in the section on DEA estimators. This estimator also allows us to

27

address the presence of undesirable outputs. Jeon and Sickles (JS) usedthis DEA estimator to examine the productivity effects of controlling forcarbon dioxide emissions on productivity growth using Mamlquist indexes.The JS estimator uses OECD data while Efficiency software uses UNIODand Bank data. The JS R codes are in a separate Jeon Sickles 2004.zipfile on the website. The default results that are printed by the R codes areproductivity growth, efficiency change, and technology change and confidenceintervals for the growth decompositions are based on the Simar and Wilson(2007) bootstrapping method. Next, we have the Simar and Zelenyuk (2006)model, which implements the Li (1996) test in the context of comparingdistributions of efficiency estimated via DEA and the Simar and Zelenyuk(2007) model that constructs confidence intervals and bias corrections forDEA-estimated aggregate efficiencies of a set of firms. It also provides a testfor the comparison of these group efficiencies.

There are other open source R code platforms other than the one thatwe have focused on in our discussions so far. The Benchmarking packagein R prepared by Peter Bogetoft and Lars Otto is one such program andthe manual and other directions for its use can be found at https://cran.r-project.org/web/packages/Benchmarking/Benchmarking.pdf. This can beused to estimate both DEA and FDH models. There have been many otherprograms developed by Simar and his colleagues in Matlab and these canbe compiled into R using Matlab-to-R converters. One such package, the‘matconv’ package by Siddarta Jairam is a useful conversion tool, althoughsome testing for accuracy is always advised for such automatic translations.8

Many of these Matlab programs can be found athttps://sites.google.com/site/productivityefficiency.9 Finally, Daraio et al.(2017a) provide a survey on a variety of software platforms that can estimateSFA and DEA models, including a number with R coding.

8See https://cran.r-project.org/web/packages/matconv/matconv.pdf for more details.9Arne Henningsen has provided the R conversion to a number of Fortran-based pro-

grams (FRONTIER) developed by Tim Coelli as well as a set of new programs in hisA Package for Stochastic Frontier Analysis (SFA) in R. This platform is available athttps://cran.r-project.org/.

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7 Summary and Final Remarks

This chapter has discussed a number of statistical and programming tech-niques used to evaluate the production efficiency of economic units, whetherthey be firms, sectors, or countries. We have discussed the methods mostwidely used for these sorts of evaluations and have provided references andurl’s to the most up to date websites that provide the R code to implementthese methods. We trust that the interested reader will find our discussionsand the software helpful for the purposes of practical evaluations of the per-formance of business entities as well as in their academic research. We havealso provided an up to date set of references that productivity and efficiencyresearchers can use for extended and deeper readings on these widely usedand adopted performance benchmarkingu; methods.

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